Convergence acceleration of logarithmic fixed point sequences

✍ Scribed by Paul Sablonniere

• Publisher: Elsevier Science
• Year: 1987
• Tongue: English
• Weight: 255 KB
• Volume: 19
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(87)80008-0

✦ Synopsis

Let (x,)

be some sequence generated by x,+ 1 = f(x,) where

i>1

For x 0 > 0 small, it converges to zero logarithmically, i.e. lim, xn+l/x . = 1, thus we need algorithms for accelerating its convergence. Using asymptotic expansions in the analysis of the A 2 and 02-algorithms leads to modified iterated versions of the first one and to combinations with the iterated 02-algorithm. In particular some superconvergence phenomena can be explained in this framework. A similar study can be made for other nonlinear algorithms known at present. Moreover, the above algorithms are also good accelerators for large classes of slowly convergent integrals and series.